[proofplan] The proof uses the defining polynomial expansion of volume under Minkowski addition. For $t \geq 0$, the volume $V(K+tL)$ is a polynomial in $t$ whose linear coefficient is $nV(K[n-1],L)$. The right derivative at $0$ is therefore obtained by subtracting the constant term and dividing by $t$, after which every term of degree at least $2$ vanishes in the limit $t \to 0+$. [/proofplan] [step:Expand the Minkowski volume as a mixed-volume polynomial] By the mixed-volume polynomial expansion for convex bodies, the function \begin{align*} f: [0,\infty) &\to \mathbb{R} \\ t &\mapsto V(K+tL) \end{align*} is given for every $t \geq 0$ by \begin{align*} f(t) = V(K+tL) = \sum_{j=0}^{n} \binom{n}{j} t^j V(K[n-j],L[j]). \end{align*} In particular, the constant term is \begin{align*} f(0)=V(K[n])=V(K), \end{align*} and the coefficient of $t$ is \begin{align*} \binom{n}{1}V(K[n-1],L)=nV(K[n-1],L). \end{align*} [/step] [step:Compute the right difference quotient] For $t>0$, subtracting the constant term from the polynomial expansion gives \begin{align*} \frac{f(t)-f(0)}{t} &= \frac{1}{t} \left( \sum_{j=0}^{n} \binom{n}{j} t^j V(K[n-j],L[j]) - V(K) \right) \\ &= \frac{1}{t} \left( \sum_{j=1}^{n} \binom{n}{j} t^j V(K[n-j],L[j]) \right) \\ &= \sum_{j=1}^{n} \binom{n}{j} t^{j-1} V(K[n-j],L[j]) \\ &= nV(K[n-1],L) + \sum_{j=2}^{n} \binom{n}{j} t^{j-1} V(K[n-j],L[j]). \end{align*} The last sum is absent when $n=1$, in which case it is interpreted as $0$. [/step] [step:Take the one-sided limit allowed by nonnegative dilations] The parameter $t$ is restricted to $[0,\infty)$ because $tL$ is the nonnegative scalar dilation of the convex body $L$. Hence the relevant derivative at $0$ is the right derivative. Since each exponent $j-1$ in the remaining sum satisfies $j-1 \geq 1$, we have \begin{align*} \lim_{t \to 0+} \sum_{j=2}^{n} \binom{n}{j} t^{j-1} V(K[n-j],L[j]) = 0. \end{align*} Therefore \begin{align*} \frac{d}{dt}\Big|_{t=0+} V(K+tL) = \lim_{t \to 0+}\frac{f(t)-f(0)}{t} = nV(K[n-1],L). \end{align*} This is the desired [first variation formula](/theorems/2728) for volume. [/step]